Given a basis $\{\gamma_{1},\ldots,\gamma_{n}\}$ of $H_{1}(F_{b},\mathbb{Z})$, then the corresponding 1-forms $\lambda_{1},\ldots,\lambda_{n}$ defined on a contractible open neighborhood $U$ of $b$ are closed and locally generate $\Lambda$. In particular, $\Lambda$ is Lagrangian with respect to the standard symplectic structure in $T^{\ast}_{B}$. A choice of functions $a_{j}$ such that $\lambda_{j}=da_{j}$ defines coordinates $a=(a_{1},\ldots,a_{n})$ called action coordinates. A covering $\{U_{i}\}$ of $B$ by contractible open sets and a choice of action coordinates on each $U_{i}$ defines an integral affine structure $\mathscr{A}$ on $B$. Moreover, if $f$ has a Lagrangian section $\sigma:U\rightarrow X$ over an open set $U\subseteq B$, then there is a natural symplectomorphism

$$\Theta:T^{\ast}_{U}/\Lambda\rightarrow f^{-1}(U).$$

(5)

If $\sigma$ is a global section then $X(B,\mathscr{A})$ is symplectically conjugate to $X$. If in addition the monodromy of $\Lambda$ is trivial $X$ is symplectically conjugate to $B\times T^{n}$. The map $\Theta$ is called the period map or action-angle coordinates map.